We don't really even know if the mother was planning eventually to get round to cleaning the seat, or whether she might have offered, or whether she was too embarrassed about it, or wasn't sure what exactly to do, or whatever - it sounds like she didn't get the chance, being taken out there in a humiliating way and told to clean the seat, in such a cold way, with her daughter standing there watching.
It's not so much saying the OP should expect to clean it up; I don't think there is necessarily that expectation, but that she could have dealt with it in so much less humiliating and patronising way.
What if it had been a guest in your house? You probably would have said don't worry, not a problem, and then they'd have offered to clean it perhaps, but if they didn't, you'd have assumed it was humiliating and they were feeling awful, and you'd have tried to make them more comfortable by pretending it didn't happen, smoothing the situation over in whatever way you could.
It sounds like you expected that if you forced her to clean it up, that 'consequence' would be enough to stop her doing it again and force her to seek treatment so that it didn't happen again. That's punishment, the same as for a small child who you are teaching, and even then, I don't think it's a very effective way of teaching. She probably has to clean herself up a lot after accidents, and that in itself isn't enough to make you seek treatment, just enough to make you depressed about it.
You don't know how many articles she might have read about it, etc. - the GP doens't have magic solutions for things like urge/spasms. I've already tried what they can offer; I knew what the physio was going to do; I'd already done most of the recommended strategies. None of it really helps, and she might be in a similar place and not really think there's much point going to the GP. The lack of belief, being told you're not trying, being told it's in your head, being told to just go home and do your pelvic floor exercises some more, the lack of understanding about just how embarrassing and anxiety-producing it is, worry about smells and stuff, all of those are reasons that I don't go back to the GP. .
She might be frightened that it's the start of the lack of control that comes with aging. She might worry it's more serious. She might be ashamed enough already. I doubt she suddenly felt that cleaning up one car seat was horrible enough that she had better seek treatment so she didn't have to do it again. If anything, she probably felt that she couldn't stand to be treated that way by someone she loved, knowing that they thought she was disgusting and lazy and always moaning but never doing anything about it, so seeking treatment might help that side of it. It might work, but it's hardly the most compassionate way of going about it.
What if the doctors can't do anything, and now she knows what you think of her and her accidents?!